Final answer:
The provided information does not clearly identify the emperor who led a successful revolt against Philip I and co-ruled with his son, as historical details about Claudius, Diocletian, and Constantine do not fully match the given description. The text also discusses the Tetrarchy and other Roman emperors without direct reference to the revolt mentioned.
Step-by-step explanation:
The emperor in question, who was declared Emperor after leading a successful revolt with his Balkan Legions against Philip I, reinstating the position of Censor, and co-ruling with his son until his death in battle fighting the Goths, is not explicitly named within the given information. However, the provided details discuss several Roman emperors and notable historical figures such as Claudius, Diocletian, Constantine, and others, as well as the establishment of the Tetrarchy, Caligula's reign, and the reigns of other emperors during the Roman Empire.
Claudius, described as the refreshingly sane emperor, was competent and oversaw the conquest of England. He was noted as a scholar and restored the imperial treasury. Although Claudius's restoration of the treasury did not directly involve his Balkan legions or a conflict with Philip I, his fiscal responsibilities and intellectual pursuits bear mentioning.
Another significant figure is Diocletian, known for his military background and major administrative reforms that included the Tetrarchy. He retired in 305 CE due to health issues. His policies had lasting effects on the Roman Empire. It's worth noting, however, that Diocletian's reign does not fully match the initial descriptions as it involves no direct conflict with Philip I or co-rule with a son until death in battle against the Goths.
Constantine, the son of Constantius Chlorus, claimed the imperial throne in 306. The details provided highlight his rise to power and the subsequent power struggles, which could be partially aligned with the description of an emperor leading successful revolts. Even though this particular narrative does not include the Battle of Milvian Bridge as a pivotal point for Constantine's ascension.